While Hamilton was at the Constitutional Convention, he gave a speech for six hours on June 19, 1787. In the speech, he called for a Constitution akin to the British model. He praised the British system of government to an extreme that his fellow New York delegates walked out on him and refused to vote the same way that he did.
His speech was a revision of the Virginia Plan. He proposed that the President should be appointed by electors, much like the Electoral College. However, he wanted the President to be appointed for life. The Congress would be composed of a House and a Senate. The House would be popularly elected, as it is now. The Senate would be elected by electors, just like the President. Also, the Senators would be appointed for life as well.
The idea is to balance the rich against the poor. The Senate and President would be more prestigious and aristocratic, whereas the House would be representative of the common man. Also, if the President were to be elected for life, he would have no reason to try and gain more power than he has, since he has a perpetual term. This would make him above corruption. The same thing applied to the Senate.
However, all Hamilton's speech did was create the idea of the Electoral College.